For \(d=2\) , formulas equivalent to the ones of Theorem REF  REF  have been first derived
in [1]}, [2]}.
The three dimensional reconstruction formulas of Theorem REF  REF  (as well as the higher dimensional generalizations of Theorem REF ) are new.
One notes, that in three spatial dimensions the reconstruction formulas are particularly
simple and further local: The reconstruction of \(f\)  at some reconstruction point \(\left(\mathbf {x}, y\right)\)  only requires the integrals over cones passing through an arbitrarily small neighbourhood of \(\left(\mathbf {x}, y\right)\) . Since for any odd \(d\) , the Riesz potential satisfies \(\mathcal {I}^{(1-d)} =\left(-1\right)^{(d-1)/2} \Delta _{\mathbf {u}}^{(d-1)/2}\) , the reconstruction formulas (REF ), ()
are in fact local for every odd space dimensions. Contrary, in even space dimension (REF ), () are
non-local: Recovering a function at a single point requires knowledge of the integrals over all conical surfaces.
This behaviour is similar to the one of the
classical Radon transform, where also the inversion is local in odd and non-local in even dimensions
(see, for example, [3]}).
